| L(s) = 1 | − 2-s − 4·5-s + 7-s + 8-s + 4·10-s + 4·11-s − 3·13-s − 14-s − 16-s − 14·17-s + 4·19-s − 4·22-s + 23-s + 5·25-s + 3·26-s − 29-s + 9·31-s + 14·34-s − 4·35-s + 4·37-s − 4·38-s − 4·40-s − 6·41-s − 11·43-s − 46-s + 6·47-s − 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.78·5-s + 0.377·7-s + 0.353·8-s + 1.26·10-s + 1.20·11-s − 0.832·13-s − 0.267·14-s − 1/4·16-s − 3.39·17-s + 0.917·19-s − 0.852·22-s + 0.208·23-s + 25-s + 0.588·26-s − 0.185·29-s + 1.61·31-s + 2.40·34-s − 0.676·35-s + 0.657·37-s − 0.648·38-s − 0.632·40-s − 0.937·41-s − 1.67·43-s − 0.147·46-s + 0.875·47-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2362886358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2362886358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.19800382182761284500871905154, −9.537154856141914626924019034323, −8.983328446436336103223658547695, −8.686338940163168085993602952564, −8.623395974887380269057849262056, −7.971331166667962069064275378578, −7.62560578139108787484362907704, −7.12475123692516587405018486480, −6.98737956701071295475766882623, −6.34275230819685207099896925689, −6.09717354140860251297452775739, −5.01379203689473519987847193449, −4.57813493017129180577693320497, −4.50873365552984938092675615328, −4.01643135865326696771623267185, −3.37378943935730196956077133266, −2.79161018573645939939805550299, −2.03530330506933544544284006759, −1.34615706998305570310593181944, −0.25340362835138770231659021176,
0.25340362835138770231659021176, 1.34615706998305570310593181944, 2.03530330506933544544284006759, 2.79161018573645939939805550299, 3.37378943935730196956077133266, 4.01643135865326696771623267185, 4.50873365552984938092675615328, 4.57813493017129180577693320497, 5.01379203689473519987847193449, 6.09717354140860251297452775739, 6.34275230819685207099896925689, 6.98737956701071295475766882623, 7.12475123692516587405018486480, 7.62560578139108787484362907704, 7.971331166667962069064275378578, 8.623395974887380269057849262056, 8.686338940163168085993602952564, 8.983328446436336103223658547695, 9.537154856141914626924019034323, 10.19800382182761284500871905154
